Like everything, it depends.
Most schools are trying to meet all their “institutional priorities” AND get some kind of SES diversity.
So if your kid is that critical pitcher or left-footed cack, it may not matter that you need FA.
If your kid looks very much like the typical FP kid but needs FA, it could be a tougher sell because they can get a version of your kid without using the FA budget.
But… many schools, having followed this pattern, have ended up with student bodies that group around both poles of the need spectrum. They are looking for kids from more middle class backgrounds who may need FA for no more than 50% of the cost. And in that group, your kid may be a standout.
It sounds like you have your head on right about this. You just need to cross your fingers and wait! It isn’t necessarily a huge disadvantage to need FA, but it can be a pretty competitive at many schools.